Inelastic cotunneling-induced decoherence and relaxation, charge, and spin currents in an interacting quantum dot under a magnetic field

We present a theoretical analysis of several aspects of nonequilibrium cotunneling through a strong Coulomb-blockaded quantum dot (QD) subject to a finite magnetic field in the weak coupling limit. We carry this out by developing a generic quantum Heisenberg-Langevin equation approach leading to a set of Bloch dynamical equations which describe the nonequilibrium cotunneling in a convenient and compact way. These equations describe the time evolution of the spin variables of the QD explicitly in terms of the response and correlation functions of the free reservoir variables. This scheme not only provides analytical expressions for the relaxation and decoherence of the localized spin induced by cotunneling, but it also facilitates evaluations of the nonequilibrium magnetization, the charge current, and the spin current at arbitrary bias-voltage, magnetic field, and temperature. We find that all cotunneling events produce decoherence, but relaxation stems only from inelastic spin-flip cotunneling processes. Moreover, our specific calculations show that cotunneling processes involving electron transfer (both spin-flip and non-spin-flip) contribute to charge current, while spin-flip cotunneling processes are required to produce a net spin current in the asymmetric coupling case. We also point out that under the influence of a nonzero magnetic field, spin-flip cotunneling is an energy-consuming process requiring a sufficiently strong external bias-voltage for activation, explaining the behavior of differential conductance at low temperature: in particular, the splitting of the zero-bias anomaly in the charge current and a broad zero-magnitude “window” of differential conductance for the spin current near zero-bias-voltage.

Figure 1

The cotunneling-induced spin relaxation rate $\left(T_1^{-1}\right)$ and decoherence rate $\left(T_2^{-1}\right)$ as functions of magnetic field $\Delta$ [(a), (b)] and of bias-voltage $V$ [(c), (d)] for given temperatures indicated in these figures. The parameters we use in calculation are $J_{LL}{\rho }_L=J_{RR}{\rho }_R=J_{LR}\sqrt{{\rho }_L{\rho }_R}=0.02$. Here, we use the decay rate of purely thermal fluctuations, ${\tau }_{\mathrm{th}}$, Eq. (29), as the unit of the two rates.

Figure 2

The cotunneling-induced spin relaxation rate $\left(T_1^{-1}\right)$ and decoherence rate $\left(T_2^{-1}\right)$ as functions of temperature for $\Delta ∕V=2.5$, 1.0, and 0. Other parameters are the same as in Fig. 1.

Figure 3

Schematic description of all 16 cotunneling processes (a)–(p) through a QD subject to Zeeman splitting energy $\Delta$ between the spin-up and -down electronic states, in the strong Coulomb blockade regime. A finite bias-voltage $V$ is applied between the two leads, $L$ and $R$. An open circle inside the QD stands for the initial occupied electron state before tunneling events, while the solid circle denotes the final state occupied by an electron after the cotunneling processes. The single-particle tunneling event ① takes place first and then is followed by the tunneling event ②, which together comprise the entire cotunneling process. The reservoir variable below each of the figures denotes the corresponding physical process occurring in the reservoir. The arrow beneath the left lead in each of the figures denotes the flow direction of the spin-up and/or spin-down electron with respect to the left lead.

Figure 4

The calculated differential conductance $d{I}^c∕dV$ vs bias-voltage $V∕\Delta$ for several temperatures at nonzero magnetic field in units of $G_0=4\pi J_{LR}^2{\rho }_L{\rho }_R$ (linear conductance at zero magnetic field). Other parameters are the same as in Fig. 1.

Figure 5

The calculated spin current, $I^s$, and its differential conductance, $d{I}^s∕dV$, as functions of bias-voltage $V∕\Delta$, at nonzero magnetic field. (a) Exhibits results for several temperatures and $J_{RR}∕J_{LL}=4.0$, $J_{LL}=0.02$. (b) Plots the results for $J_{RR}∕J_{LL}=5.0$ $(J_{LL}=0.02)$ as solid lines, and for $J_{LL}∕J_{RR}=5.0$ $(J_{RR}=0.02)$ as dashed lines.